Harvey Friedman writes:
> When I see this Paradox discussed, I always want to know what some
> criteria are for a "solution". I wish scholars would pay more
> attention to this issue.
I am looking for an algorithm that assigns descriptions to sentences
that themselves describe other sentences. The descriptions considered
include describing sentences as "true"; however, if the algorithm
describes sentences in some other way, such as "paradoxical", then it
must be applicable to sentences that themselves describe other sentences
as paradoxical.
An ideal, surely unobtainable, would be if a sentence could be called
true by the algorithm if, and only if, it described sentences in the
same way the algorithm described those sentences. This is seemingly
impossible, since the strong Liar is presumably to be described as
"paradoxical", and the strong Liar describes itself as paradoxical,
nevertheless it is not described as true. But the algorithm could at
least meet this criterion:
1. Any sentence that is not paradoxical will be called true if, and only
if, it describes sentences consistent with way the algorithm describes
those sentences. (and false it describes sentences in a way
inconsistent).
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The idea behind the second criterion is that the algorithm should call
its own conclusions, true. But (1) will guarantee this, provided the
conclusions can be expressed in sentences that are not paradoxical.
An ideal case would be, that whatever we want to say about a sentence,
we can construct a non-paradoxical sentence that says it. But Alan
Weir claims to have found a sentence such that if we try to call it not
true, our sentence will be referred to by the sentence itself, in a way
that creates a paradox. If this example is indeed valid, the algorithm
may say of Weir's sentence that it does not in fact succeed in referring
to those sentences to which it seems to refer. This would be a
limitation on the ability of a sentence W to refer to the set of
sentences that call W true. Such a limitation on reference is
unfortunate: the best we can hope for is that such cases (if they
exist) are rare, so the sentences we need for both ordinary logic and
everyday life can be described by the algorithm, and the algorithm can
call its own conclusions about them "true".
In short, I don't know what to make of Weir's example. But I don't
accept it as an argument that its OK to describe sentences using an
algorithm that can't call ANY of its conclusions true.
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The third criterion is that the algorithm should not call too many
sentences paradoxical. For any sentence called paradoxical, there
should be a reason why it can't be called true, and can't be called
false.
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I have an algorithm which I think has some interesting properties, but
I'm not sure if details of particular algorithms are of interest.
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Sandy Hodges / Alameda, California, USA
mail to SandyHodges at attbi.com will reach me.